Description: The GLB of the inclusion poset. (hypotheses "ipolub.s" and "ipoglb.t" could be eliminated with S e. dom G .) Could be significantly shortened if posglbdg is in quantified form. mrelatglb could potentially be shortened using this. See mrelatglbALT . (Contributed by Zhi Wang, 29-Sep-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ipolub.i | |
|
ipolub.f | |
||
ipolub.s | |
||
ipoglb.g | |
||
ipoglbdm.t | |
||
ipoglb.t | |
||
Assertion | ipoglb | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipolub.i | |
|
2 | ipolub.f | |
|
3 | ipolub.s | |
|
4 | ipoglb.g | |
|
5 | ipoglbdm.t | |
|
6 | ipoglb.t | |
|
7 | eqid | |
|
8 | 1 | ipobas | |
9 | 2 8 | syl | |
10 | 1 | ipopos | |
11 | 10 | a1i | |
12 | breq2 | |
|
13 | unilbeu | |
|
14 | 13 | biimpar | |
15 | 6 5 14 | syl2anc | |
16 | 1 2 3 7 | ipoglblem | |
17 | 6 16 | mpdan | |
18 | 15 17 | mpbid | |
19 | 18 | simpld | |
20 | 19 | adantr | |
21 | simpr | |
|
22 | 12 20 21 | rspcdva | |
23 | breq1 | |
|
24 | 23 | ralbidv | |
25 | breq2 | |
|
26 | 25 | cbvralvw | |
27 | 24 26 | bitrdi | |
28 | breq1 | |
|
29 | 27 28 | imbi12d | |
30 | 18 | simprd | |
31 | 30 | adantr | |
32 | simpr | |
|
33 | 29 31 32 | rspcdva | |
34 | 33 | 3impia | |
35 | 7 9 4 11 3 6 22 34 | posglbdg | |